
Assume that \(\mathbb {T}_p\)  is a \((p+1)\) -dimensional ECT-space on the interval \([a,b]\)  and it contains constants. Moreover, assume that the derivative space of \(\mathbb {T}_p\)  is a \(p\) -dimensional ECT-space on \([a,b]\)  generated by the weights \(\lbrace  w_1,\ldots ,w_p\rbrace \) .
If all the spaces \(\mathbb {T}_{p,i}\) , \(i=1,\ldots , m\)  in (REF ) are selected as the restriction of \(\mathbb {T}_p\) , then from rmkgen-powers-der it follows that the weights \(\lbrace 1,w_1,\ldots ,w_p\rbrace \) 
restricted to the intervals \([x_{i-1}, x_i]\) , \(i=1,\ldots , m\)  form admissible weight systems for the space \(\mathbb {S}_p^{\mathbf {r}}(\Delta )\) .
In other words, if all the pieces of the Tchebycheffian spline functions are taken from a single space \(\mathbb {T}_p\)  which is an ECT-space on \([a,b]\)  containing constants, then the space \(\mathbb {S}_p^{\mathbf {r}}(\Delta )\)  admits a TB-spline basis provided that the derivative space of \(\mathbb {T}_p\)  is also an ECT-space on \([a,b]\) ; see [1]}, [2]}, [3]} and references therein.

Assume now that \(\mathbb {T}_p\)  is a \((p+1)\) -dimensional ECT-space on each interval \([x_{i-1}, x_i]\)  separately, but not necessarily on the entire interval \([a,b]\) .
Moreover, assume that \(\mathbb {T}_p\)  is the null-space of a linear differential operator as in (REF ) and it contains constants.
If all the pieces of the Tchebycheffian spline functions are taken from such space \(\mathbb {T}_p\) , then the space \(\mathbb {S}_p^{\mathbf {r}}(\Delta )\)  admits a TB-spline basis on condition that
\(h_\Delta \max _{i=1,\ldots ,m}(x_{i}-x_{i-1})\)  is sufficiently small. In particular, it admits a TB-spline basis when the derivative space of \(\mathbb {T}_p\)  is an ECT-space on each interval \([\xi _{k+1},\xi _{k+p}]\) , where \(\xi _{k+1}<\xi _{k+p}\)  and \(1\le k\le n\) . We refer the reader to [4]} for details (and also [5]}). Note that these intervals are closely related to the supports of the TB-splines; see Proposition REF .

Admissible weight systems can be easily obtained for the interesting subclass of generalized polynomial splines, i.e., splines where the pieces are taken from generalized polynomial spaces (such as (REF ) and (REF )), even considering different spaces on different intervals. This ensures the existence of TB-splines, called generalized polynomial B-splines (GB-splines) in this setting, under very mild restrictions. In particular, the space \(\mathbb {S}_p^{\mathbf {r}}(\Delta )\)  admits a TB-spline basis when the \((p-1)\) -th order derivative space of \(\mathbb {T}_{p,i}\)  in (REF ) is an ECT-space on \([x_{i-1},x_{i}]\) , \(i=1,\ldots , m\) . We refer the reader to [1]} for details.

